Significant figures are an expression of the uncertainty of a number. All measurements have some degree of uncertainty in their value. This can be caused by the skill of the measurement taker or the tool used.

For example, if you are in a chemistry lab and you need to add 8 mL of liquid to a beaker. You could just pour water straight into the beaker and quit when you think you hit 8 mL. The error associated with this measurement would mostly be associated to your skill. You could use a beaker with markings every 5 mL and get pretty close, give or take a couple mL. You could use a graduated cylinder with markings every tenth of a mL and reliably get measurements between 7.9 and 8.1 mL. Here we see how the uncertainty can be affected by the measuring tool.

### Significant Figure Rules

Uncertainty is expressed in significant figures. The more significant figures in a measurement, the more precise the measurement. There are six basic rules dealing with significant figures.

- Non-zero digits are always significant.
- All zeros between other significant digits are significant.
- The most significant figure, also called most significant digit, is the leftmost non-zero digit. For example: in the number 0.00321, the most significant figure is the 3.
- The least significant figure, or least significant digit is the rightmost digit. In the number 54.321, the least significant figure is 1. Keep in mind, zero can be the least significant digit. For example, the zero in 4.320 is the least significant figure.
- Any zero digit to the right of the decimal point is significant.

For example 2 has one significant digit, but 2.0 has two significant figures. - If no decimal point is present, the rightmost non-zero digit is the least significant figure.

**Quick Tip to Calculate Significant Figures**

Write the number in scientific notation. The numbers ahead of the multiplier are all significant.

Example: How many significant figures are in the following numbers?

a) 23,000

b) 0.000504

c) 240.05

d) 4.000

Write each number in scientific notation.

a) 2.3 x 10^{3}

b)5.04 x 10^{-4}

c) 2.4005 x10^{2}

d) 4.000 x 10^{1}

Now we can easily count the digits ahead of the multiplier to get the number of significant figures.

a) 2 significant figures

b) 3 significant figures

c) 5 significant figures

d) 4 significant figures

### Significant Figures and Uncertainty in Calculations

Once you have your measurement, you are most likely going to use it in a calculation of some sort. In a calculation, the uncertainty of the result is determined by the uncertainty of the measurements.

**Addition and Subtraction**

When measurements are used in addition and subtraction, the uncertainty is determined by the uncertainty of the least precise measurement, not by the number of significant figures.

Example: Add the following three measurements: 24.21 cm, 5.005 cm and 22 cm.

If you add them up, you get 51.215 m. The least precise measurement is the 22 cm measurement, so the answer should have the same precision.

The value of the calculation would be reported as 51 m.

**Multiplication and Division**

When measurements are used in multiplication and division, the number of significant figures in the result with be the same as the number with the smallest number of significant figures.**Example:** Divide 35.105 grams by 35 mL.

If you just divide the two numbers, you get 1.003 g/mL. The value you would report depends on the measurement with the least significant figures. The first measurement has 5 significant and the second has only 2 significant figures.

The reported value would then be 1.0 g/mL

**Losing Significant Figures**

Significant figures can be ‘lost’ in a calculation. For example, if you have a beaker that weighs 75.206 grams and you add water until the weight is 75.844. The water would weigh the difference between these two values.

75.844 g – 75.206 g = 0.638 g

The final result only has 3 significant figures when both measurements had 5 significant figures.

**Exact Numbers**

Occasionally, a calculation involves a number with an exact value rather than an approximation. This occurs in calculations using conversion factors, pure numbers or physical constants. The significant figures of these numbers do not affect the end result. For example, if you were to find the average of 10.3 cm, 12.7 cm and 14.5 cm, you would add the three numbers together to get 37.5 cm. You would then divide this by 3 to get the average or 12.5 cm. Even though 3 only has one significant figure, your answer is still 12.5 cm.

The use and rules of significant figures in science and engineering is standard in any field. Measuring is a basic skill in science and everyone needs to work under the same rules. It is best to learn them early and keep them in mind in all your work.

**More about Measurements:**

What is the Difference Between Accuracy and Precision?